Mandelbrot Boundary Area












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$begingroup$


I might be entirely off an my assumptions, but the following has led me to a question.




  1. The Mandelbrot set is contained by an border of infinite length.


  2. Said border is 2-dimensional.


  3. The Hilbert space-filling curve also is 2-dimensional and infinite in length.


  4. The Hilbert curve covers an area of 1 square unit.



Can the Mandelbrot border be assigned an area? Am I justified in saying that the Hilbert curve is has area 1? If it is two dimensional, that seems reasonable. What terminology is used for non-integer dimensions? Also, for integer dimensions. What do you call the measure of the 4D interior of a tesseract? 5D?



I'm in 10th grade, but very good at math (taking Calc BC), so as long as answers don't go to deep into metric spaces, measure theory, topology, et cetera, I should be able to follow them.



Thanks in advance!










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$endgroup$












  • $begingroup$
    See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
    $endgroup$
    – lhf
    Dec 25 '18 at 17:20


















1












$begingroup$


I might be entirely off an my assumptions, but the following has led me to a question.




  1. The Mandelbrot set is contained by an border of infinite length.


  2. Said border is 2-dimensional.


  3. The Hilbert space-filling curve also is 2-dimensional and infinite in length.


  4. The Hilbert curve covers an area of 1 square unit.



Can the Mandelbrot border be assigned an area? Am I justified in saying that the Hilbert curve is has area 1? If it is two dimensional, that seems reasonable. What terminology is used for non-integer dimensions? Also, for integer dimensions. What do you call the measure of the 4D interior of a tesseract? 5D?



I'm in 10th grade, but very good at math (taking Calc BC), so as long as answers don't go to deep into metric spaces, measure theory, topology, et cetera, I should be able to follow them.



Thanks in advance!










share|cite|improve this question









$endgroup$












  • $begingroup$
    See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
    $endgroup$
    – lhf
    Dec 25 '18 at 17:20
















1












1








1


1



$begingroup$


I might be entirely off an my assumptions, but the following has led me to a question.




  1. The Mandelbrot set is contained by an border of infinite length.


  2. Said border is 2-dimensional.


  3. The Hilbert space-filling curve also is 2-dimensional and infinite in length.


  4. The Hilbert curve covers an area of 1 square unit.



Can the Mandelbrot border be assigned an area? Am I justified in saying that the Hilbert curve is has area 1? If it is two dimensional, that seems reasonable. What terminology is used for non-integer dimensions? Also, for integer dimensions. What do you call the measure of the 4D interior of a tesseract? 5D?



I'm in 10th grade, but very good at math (taking Calc BC), so as long as answers don't go to deep into metric spaces, measure theory, topology, et cetera, I should be able to follow them.



Thanks in advance!










share|cite|improve this question









$endgroup$




I might be entirely off an my assumptions, but the following has led me to a question.




  1. The Mandelbrot set is contained by an border of infinite length.


  2. Said border is 2-dimensional.


  3. The Hilbert space-filling curve also is 2-dimensional and infinite in length.


  4. The Hilbert curve covers an area of 1 square unit.



Can the Mandelbrot border be assigned an area? Am I justified in saying that the Hilbert curve is has area 1? If it is two dimensional, that seems reasonable. What terminology is used for non-integer dimensions? Also, for integer dimensions. What do you call the measure of the 4D interior of a tesseract? 5D?



I'm in 10th grade, but very good at math (taking Calc BC), so as long as answers don't go to deep into metric spaces, measure theory, topology, et cetera, I should be able to follow them.



Thanks in advance!







measure-theory fractals dimension-theory






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share|cite|improve this question











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share|cite|improve this question










asked Dec 25 '18 at 15:40









William GrannisWilliam Grannis

998521




998521












  • $begingroup$
    See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
    $endgroup$
    – lhf
    Dec 25 '18 at 17:20




















  • $begingroup$
    See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
    $endgroup$
    – lhf
    Dec 25 '18 at 17:20


















$begingroup$
See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
$endgroup$
– lhf
Dec 25 '18 at 17:20






$begingroup$
See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
$endgroup$
– lhf
Dec 25 '18 at 17:20












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